Sparse and Greedy: Sparsifying Submodular Facility Location Problems

Submodular facility location functions are widely used for summarizing large datasets and have found applications ranging from sensor placement, image retrieval, and clustering. A significant problem is that evaluating such functions typically requires the calculation of pairwise benefits for all items, which is computationally unmanageable for large problems. In this paper we propose a sparsification method that only computes nearest-neighbor benefits and can dramatically accelerate submodular greedy optimization. We show that the optimal solution of the sparsified problem will be at most a factor of a half from optimal, under minimal assumptions.